Newform 4005.2.a.p

• Label: 4005.2.a.p
• Level: 4005
• Weight: 2
• Character orbit: 4005.a
• Self dual: Yes
• Analytic conductor: 31.980
• Analytic rank: 1
• Dimension: 8
• CM: No
• Inner twists: 1

Newspace parameters

Level:	\( N \)	=	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	4005.a (trivial)

Newform invariants

Self dual:	Yes
Analytic conductor:	\(31.9800860095\)
Analytic rank:	\(1\)
Dimension:	\(8\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2 \)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q -\beta_{1} q^{2} + ( 1 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{4} + q^{5} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{7} + ( -\beta_{1} - \beta_{2} ) q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8q - q^{2} + 7q^{4} + 8q^{5} - 6q^{7} - 3q^{8} + O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 11 x^{6} + 9 x^{5} + 34 x^{4} - 19 x^{3} - 27 x^{2} + 11 x - 1\):

\(\beta_{0}\)	\(=\)	\( 1 \)
\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{3} - 5 \nu \)
\(\beta_{3}\)	\(=\)	\((\)\( -2 \nu^{7} + \nu^{6} + 23 \nu^{5} - 8 \nu^{4} - 76 \nu^{3} + 12 \nu^{2} + 65 \nu - 7 \)\()/2\)
\(\beta_{4}\)	\(=\)	\( -\nu^{7} + \nu^{6} + 11 \nu^{5} - 9 \nu^{4} - 34 \nu^{3} + 18 \nu^{2} + 27 \nu - 7 \)
\(\beta_{5}\)	\(=\)	\((\)\( -4 \nu^{7} + 3 \nu^{6} + 45 \nu^{5} - 26 \nu^{4} - 144 \nu^{3} + 50 \nu^{2} + 121 \nu - 27 \)\()/2\)
\(\beta_{6}\)	\(=\)	\((\)\( 5 \nu^{7} - 4 \nu^{6} - 55 \nu^{5} + 34 \nu^{4} + 170 \nu^{3} - 61 \nu^{2} - 136 \nu + 27 \)\()/2\)
\(\beta_{7}\)	\(=\)	\((\)\( -6 \nu^{7} + 5 \nu^{6} + 67 \nu^{5} - 42 \nu^{4} - 212 \nu^{3} + 72 \nu^{2} + 175 \nu - 29 \)\()/2\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

2.51468
2.23321
1.16343
0.217002
0.151894
−1.11667
−1.66289
−2.50065

−2.51468

0

4.32363

1.00000

0

−2.83060

−5.84320

0

−2.51468

1.2

−2.23321

0

2.98721

1.00000

0

1.87039

−2.20463

0

−2.23321

1.3

−1.16343

0

−0.646427

1.00000

0

2.23155

3.07894

0

−1.16343

1.4

−0.217002

0

−1.95291

1.00000

0

−2.89958

0.857790

0

−0.217002

1.5

−0.151894

0

−1.97693

1.00000

0

2.88745

0.604071

0

−0.151894

1.6

1.11667

0

−0.753041

1.00000

0

−1.16728

−3.07425

0

1.11667

1.7

1.66289

0

0.765209

1.00000

0

−1.19579

−2.05332

0

1.66289

1.8

2.50065

0

4.25326

1.00000

0

−4.89614

5.63461

0

2.50065

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(5\)	\(-1\)
\(89\)	\(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\(T_{2}^{8} + \cdots\)

\(T_{7}^{8} + \cdots\)

\(T_{11}^{8} + \cdots\)